Fermat point

The Fermat point (F) is the point inside a triangle ABC (provided no angle exceeds 120 degrees) such that the sum AF + BF + CF is a minimum. According to a theorem (generally attributed to Napoleon Bonaparte), the circumcircles of equilateral triangle constructed on the outside of triangle ABC meet at the Fermat point and their centres form a fourth equilateral triangle - the Napoleon Triangle.

For a discussion on the existence of this Fermat point click below:
1. Discussion on the intersection of the circumcircles,
2. Discussion illustrating that AF + BF + CF is a minimum.